Comments

Okay brother the solution is simple
First find the no of solutions without any restriction
Now you have to subtract 3 cases,
Case 1
When a =5
then the solve b+c+d=5;
Case 2
When b=4;
Case 3
When both a and b are at their maximas.;
Subtract these cases from your solution.
PS- case 3 is not valid in this case.